In mathematics, the Newton polygon is a tool for understanding the behaviour of polynomials over local fields, or more generally, over ultrametric fields. In the original case, the local field of interest was essentially the field of formal Laurent series in the indeterminate X, i.e. the field of fractions of the formal power series ring K [ [ X ] ] {\displaystyle K[[X]]} , over K {\displaystyle K} , where K {\displaystyle K} was the real number or complex number field. This is still of considerable utility with respect to Puiseux expansions. The Newton polygon is an effective device for understanding the leading terms a X r {\displaystyle aX^{r}} of the power series expansion solutions to equations P ( F ( X ) ) = 0 {\displaystyle P(F(X))=0} where P {\displaystyle P} is a polynomial with coefficients in K [ X ] {\displaystyle K[X]} , the polynomial ring; that is, implicitly defined algebraic functions. The exponents r {\displaystyle r} here are certain rational numbers, depending on the branch chosen; and the solutions themselves are power series in K [ [ Y ] ] {\displaystyle K[[Y]]} with Y = X 1 d {\displaystyle Y=X^{\frac {1}{d}}} for a denominator d {\displaystyle d} corresponding to the branch. The Newton polygon gives an effective, algorithmic approach to calculating d {\displaystyle d} .
After the introduction of the p-adic numbers, it was shown that the Newton polygon is just as useful in questions of ramification for local fields, and hence in algebraic number theory. Newton polygons have also been useful in the study of elliptic curves.
A priori, given a polynomial over a field, the behaviour of the roots (assuming it has roots) will be unknown. Newton polygons provide one technique for the study of the behaviour of the roots.
Let K {\displaystyle K} be a field endowed with a non-archimedean valuation v K : K → → --> R ∪ ∪ --> { ∞ ∞ --> } {\displaystyle v_{K}:K\to \mathbb {R} \cup \{\infty \}} , and let
with a 0 a n ≠ ≠ --> 0 {\displaystyle a_{0}a_{n}\neq 0} . Then the Newton polygon of f {\displaystyle f} is defined to be the lower boundary of the convex hull of the set of points P i = ( i , v K ( a i ) ) , {\displaystyle P_{i}=\left(i,v_{K}(a_{i})\right),} ignoring the points with a i = 0 {\displaystyle a_{i}=0} .
Restated geometrically, plot all of these points Pi on the xy-plane. Let's assume that the points indices increase from left to right (P0 is the leftmost point, Pn is the rightmost point). Then, starting at P0, draw a ray straight down parallel with the y-axis, and rotate this ray counter-clockwise until it hits the point Pk1 (not necessarily P1). Break the ray here. Now draw a second ray from Pk1 straight down parallel with the y-axis, and rotate this ray counter-clockwise until it hits the point Pk2. Continue until the process reaches the point Pn; the resulting polygon (containing the points P0, Pk1, Pk2, ..., Pkm, Pn) is the Newton polygon.
Another, perhaps more intuitive way to view this process is this : consider a rubber band surrounding all the points P0, ..., Pn. Stretch the band upwards, such that the band is stuck on its lower side by some of the points (the points act like nails, partially hammered into the xy plane). The vertices of the Newton polygon are exactly those points.
For a neat diagram of this see Ch6 §3 of "Local Fields" by JWS Cassels, LMS Student Texts 3, CUP 1986. It is on p99 of the 1986 paperback edition.
With the notations in the previous section, the main result concerning the Newton polygon is the following theorem,[1] which states that the valuation of the roots of f {\displaystyle f} are entirely determined by its Newton polygon:
Let μ μ --> 1 , μ μ --> 2 , … … --> , μ μ --> r {\displaystyle \mu _{1},\mu _{2},\ldots ,\mu _{r}} be the slopes of the line segments of the Newton polygon of f ( x ) {\displaystyle f(x)} (as defined above) arranged in increasing order, and let λ λ --> 1 , λ λ --> 2 , … … --> , λ λ --> r {\displaystyle \lambda _{1},\lambda _{2},\ldots ,\lambda _{r}} be the corresponding lengths of the line segments projected onto the x-axis (i.e. if we have a line segment stretching between the points P i {\displaystyle P_{i}} and P j {\displaystyle P_{j}} then the length is j − − --> i {\displaystyle j-i} ).
With the notation of the previous sections, we denote, in what follows, by L {\displaystyle L} the splitting field of f {\displaystyle f} over K {\displaystyle K} , and by v L {\displaystyle v_{L}} an extension of v K {\displaystyle v_{K}} to L {\displaystyle L} .
Newton polygon theorem is often used to show the irreducibility of polynomials, as in the next corollary for example:
Indeed, by the main theorem, if α α --> {\displaystyle \alpha } is a root of f {\displaystyle f} , v L ( α α --> ) = − − --> a / n . {\displaystyle v_{L}(\alpha )=-a/n.} If f {\displaystyle f} were not irreducible over K {\displaystyle K} , then the degree d {\displaystyle d} of α α --> {\displaystyle \alpha } would be < n {\displaystyle <n} , and there would hold v L ( α α --> ) ∈ ∈ --> 1 d Z {\displaystyle v_{L}(\alpha )\in {1 \over d}\mathbb {Z} } . But this is impossible since v L ( α α --> ) = − − --> a / n {\displaystyle v_{L}(\alpha )=-a/n} with a {\displaystyle a} coprime to n {\displaystyle n} .
Another simple corollary is the following:
Proof: By the main theorem, f {\displaystyle f} must have a single root α α --> {\displaystyle \alpha } whose valuation is v L ( α α --> ) = − − --> μ μ --> i . {\displaystyle v_{L}(\alpha )=-\mu _{i}.} In particular, α α --> {\displaystyle \alpha } is separable over K {\displaystyle K} . If α α --> {\displaystyle \alpha } does not belong to K {\displaystyle K} , α α --> {\displaystyle \alpha } has a distinct Galois conjugate α α --> ′ {\displaystyle \alpha '} over K {\displaystyle K} , with v L ( α α --> ′ ) = v L ( α α --> ) {\displaystyle v_{L}(\alpha ')=v_{L}(\alpha )} ,[2] and α α --> ′ {\displaystyle \alpha '} is a root of f {\displaystyle f} , a contradiction.
More generally, the following factorization theorem holds:
Proof: For every i {\displaystyle i} , denote by f i {\displaystyle f_{i}} the product of the monomials ( X − − --> α α --> ) {\displaystyle (X-\alpha )} such that α α --> {\displaystyle \alpha } is a root of f {\displaystyle f} and v L ( α α --> ) = − − --> μ μ --> i {\displaystyle v_{L}(\alpha )=-\mu _{i}} . We also denote f = A P 1 k 1 P 2 k 2 ⋯ ⋯ --> P s k s {\displaystyle f=AP_{1}^{k_{1}}P_{2}^{k_{2}}\cdots P_{s}^{k_{s}}} the factorization of f {\displaystyle f} in K [ X ] {\displaystyle K[X]} into prime monic factors ( A ∈ ∈ --> K ) . {\displaystyle (A\in K).} Let α α --> {\displaystyle \alpha } be a root of f i {\displaystyle f_{i}} . We can assume that P 1 {\displaystyle P_{1}} is the minimal polynomial of α α --> {\displaystyle \alpha } over K {\displaystyle K} . If α α --> ′ {\displaystyle \alpha '} is a root of P 1 {\displaystyle P_{1}} , there exists a K-automorphism σ σ --> {\displaystyle \sigma } of L {\displaystyle L} that sends α α --> {\displaystyle \alpha } to α α --> ′ {\displaystyle \alpha '} , and we have v L ( σ σ --> α α --> ) = v L ( α α --> ) {\displaystyle v_{L}(\sigma \alpha )=v_{L}(\alpha )} since K {\displaystyle K} is Henselian. Therefore α α --> ′ {\displaystyle \alpha '} is also a root of f i {\displaystyle f_{i}} . Moreover, every root of P 1 {\displaystyle P_{1}} of multiplicity ν ν --> {\displaystyle \nu } is clearly a root of f i {\displaystyle f_{i}} of multiplicity k 1 ν ν --> {\displaystyle k_{1}\nu } , since repeated roots share obviously the same valuation. This shows that P 1 k 1 {\displaystyle P_{1}^{k_{1}}} divides f i . {\displaystyle f_{i}.} Let g i = f i / P 1 k 1 {\displaystyle g_{i}=f_{i}/P_{1}^{k_{1}}} . Choose a root β β --> {\displaystyle \beta } of g i {\displaystyle g_{i}} . Notice that the roots of g i {\displaystyle g_{i}} are distinct from the roots of P 1 {\displaystyle P_{1}} . Repeat the previous argument with the minimal polynomial of β β --> {\displaystyle \beta } over K {\displaystyle K} , assumed w.l.g. to be P 2 {\displaystyle P_{2}} , to show that P 2 k 2 {\displaystyle P_{2}^{k_{2}}} divides g i {\displaystyle g_{i}} . Continuing this process until all the roots of f i {\displaystyle f_{i}} are exhausted, one eventually arrives to f i = P 1 k 1 ⋯ ⋯ --> P m k m {\displaystyle f_{i}=P_{1}^{k_{1}}\cdots P_{m}^{k_{m}}} , with m ≤ ≤ --> s {\displaystyle m\leq s} . This shows that f i ∈ ∈ --> K [ X ] {\displaystyle f_{i}\in K[X]} , f i {\displaystyle f_{i}} monic. But the f i {\displaystyle f_{i}} are coprime since their roots have distinct valuations. Hence clearly f = A f 1 ⋅ ⋅ --> f 2 ⋯ ⋯ --> f r {\displaystyle f=Af_{1}\cdot f_{2}\cdots f_{r}} , showing the main contention. The fact that λ λ --> i = deg --> ( f i ) {\displaystyle \lambda _{i}=\deg(f_{i})} follows from the main theorem, and so does the fact that μ μ --> i = v K ( f i ( 0 ) ) / λ λ --> i {\displaystyle \mu _{i}=v_{K}(f_{i}(0))/\lambda _{i}} , by remarking that the Newton polygon of f i {\displaystyle f_{i}} can have only one segment joining ( 0 , v K ( f i ( 0 ) ) {\displaystyle (0,v_{K}(f_{i}(0))} to ( λ λ --> i , 0 = v K ( 1 ) ) {\displaystyle (\lambda _{i},0=v_{K}(1))} . The condition for the irreducibility of f i {\displaystyle f_{i}} follows from the corollary above. (q.e.d.)
The following is an immediate corollary of the factorization above, and constitutes a test for the reducibility of polynomials over Henselian fields:
Other applications of the Newton polygon comes from the fact that a Newton Polygon is sometimes a special case of a Newton polytope, and can be used to construct asymptotic solutions of two-variable polynomial equations like 3 x 2 y 3 − − --> x y 2 + 2 x 2 y 2 − − --> x 3 y = 0. {\displaystyle 3x^{2}y^{3}-xy^{2}+2x^{2}y^{2}-x^{3}y=0.}
In the context of a valuation, we are given certain information in the form of the valuations of elementary symmetric functions of the roots of a polynomial, and require information on the valuations of the actual roots, in an algebraic closure. This has aspects both of ramification theory and singularity theory. The valid inferences possible are to the valuations of power sums, by means of Newton's identities.
Newton polygons are named after Isaac Newton, who first described them and some of their uses in correspondence from the year 1676 addressed to Henry Oldenburg.[4]
Museo de Antropología de Xalapa Cabeza olmeca, San Lorenzo Tenochtitlán, vista de frente y perfilUbicaciónPaís México MéxicoLocalidad Xalapa-EnriquezCoordenadas 19°33′02″N 96°55′52″O / 19.5505, -96.931Tipo y coleccionesTipo AntropológiaHistoria y gestiónInauguración 1957Administrador C.P. Jesús Armando Saint Martin ContrerasDirector Maura Ordóñez ValenzuelaInformación para visitantesVisitantes Dato pendiente[editar datos en Wikidata] El Museo de A…
هذه المقالة يتيمة إذ تصل إليها مقالات أخرى قليلة جدًا. فضلًا، ساعد بإضافة وصلة إليها في مقالات متعلقة بها. (أكتوبر 2015) الأسرار المقدسة (بالإنجليزية Mysterium) عمل موسيقي لم يكتمل للمؤلف الموسيقي أليكساندر سكريابين بدأ العمل في كتابته عام 1903، لكن لم يكمله عند وفاته عام 1915. خطط سكري
American musician and filmmaker Matt FarleyFarley in 2019Background informationBorn (1978-06-03) June 3, 1978 (age 45)OriginDanvers, Massachusetts, U.S.GenresAlternative rock, rock music, folk, novelty songsOccupation(s)Singer-songwriter, musician, filmmakerInstrument(s)Vocals, piano, keyboards, guitarYears active1996–presentLabelsMotern MediaWebsitemoternmedia.comMusical artist Matt Farley (born June 3, 1978)[1] is an American filmmaker, musician, and songwriter who has released …
「苑里」重定向至此。关于其他用法,请见「苑裡 (消歧义)」。 苑裡鎮舊稱:宛里社鎮 苑裡鎮位置圖 坐标:24°25′N 120°41′E / 24.42°N 120.68°E / 24.42; 120.68國家 中華民國省臺灣省(已虛級化)上級區劃苗栗縣下級區劃25里360鄰政府 • 行政机构苑裡鎮公所(立法機關:苑裡鎮民代表會) • 鎮長劉育育(苑裡鎮鎮長列表)面积 •&…
Pour les articles homonymes, voir Hédon. Emmanuel HédonBiographieNaissance 30 avril 1863BurieDécès 8 mars 1933 (à 69 ans)MontpellierNom de naissance Charles Édouard Eutrope Emmanuel HédonNationalité françaiseFormation Université de BordeauxFaculté de médecine de MontpellierActivités Médecin, physiologisteAutres informationsA travaillé pour Faculté de médecine de MontpellierMembre de Académie nationale de médecineAcadémie des sciences et lettres de MontpellierSociété de…
American science fiction author (1925–2012) Harry HarrisonHarrison in 2005BornHenry Maxwell Dempsey(1925-03-12)March 12, 1925Stamford, Connecticut, U.S.DiedAugust 15, 2012(2012-08-15) (aged 87)Brighton, EnglandOccupationWriter, illustratorNationalityAmerican, IrishPeriod1951–2010GenreScience fiction, satireNotable awardsInkpot Award (2004)[1]SpouseEvelyn Harrison (div. 1951)Joan Merkler Harrison (1954–2002, her death)Children2Websiteharryharrison.com Harry Max Harrison (born H…
Data center monitoring system for operating systems System Center Operations ManagerDeveloper(s)MicrosoftStable release2022 UR1 / 14 December 2022 Operating systemMicrosoft WindowsTypeNetwork administration System monitorLicenseTrialwareWebsitedocs.microsoft.com/en-us/system-center/scom/ System Center Operations Manager (SCOM) is a cross-platform data center monitoring system for operating systems and hypervisors. It uses a single interface that shows state, health, and performance information o…
Fetishization of Black culture For the album, see Negrophilia (album). Not to be confused with Necrophilia, the act of having sex with a corpse. Josephine Baker dancing the Charleston at the Folies Bergère, Paris, in 1926 Nancy Cunard (1928), activist, heiress and negrophile, with an unidentified partner Josephine Baker in her famous skirt of bananas during her performance in La Folie du JourThe word negrophilia[1] is derived from the French négrophilie that means love of the Negro. …
Keuskupan Agung Feira de SantanaArchidioecesis Fori Sancti AnnaeArquidiocese de Feira de SantanaKatedral Metropolitan Santa AnnaLokasiNegara BrazilProvinsi gerejawiFeira de SantanaStatistikLuas16.878 km2 (6.517 sq mi)Populasi- Total- Katolik(per 2006)960.033859,949 (89.6%)InformasiRitusRitus LatinPendirian21 Juli 1962 (61 tahun lalu)KatedralKatedral Santa Anna di Feira de SantanaKepemimpinan kiniPausFransiskusUskup agungZanoni Demettino CastroSitus webwww.…
Cet article est une ébauche concernant un canton français et les Alpes-de-Haute-Provence. Vous pouvez partager vos connaissances en l’améliorant (comment ?) selon les recommandations des projets correspondants. Canton de Manosque-Sud-Est Administration Pays France Région Provence-Alpes-Côte d'Azur Département Alpes-de-Haute-Provence Arrondissement(s) Forcalquier Chef-lieu Manosque Conseiller général Mandat Yannick Philipponneau 2011-2015 Code canton 04 32 Disparition 29 mars 2015[…
YF-17 CobraYF-17 Cobra terbang diatas gurun pasir.TipePrototipTerbang perdana9 Juni 1974Pengguna utamaAmerika SerikatJumlah produksi2Harga satuanUS$28 Juta (Rp433,49 Miliar)Acuan dasarNorthrop F-5[1]VarianF/A-18 Hornet YF-17 Cobra merupakan prototip pesawat tempur siang hari ringan yang dirancang untuk program Light Weight Fighter (LWF, Pesawat Tempur Ringan) Angkatan Udara Amerika Serikat. Program LWF diadakan karena komunitas pesawat tempur Amerika Serikat beranggapan bahwa F-15 Eagle …
BMW Seri 6 (E63)InformasiProdusenBMWMasa produksi2003–2010PerakitanDingolfing, Jerman6th of October City, Mesir(BAG)PerancangAdrian van Hooydonk (2000)Bodi & rangkaBentuk kerangka2-pintu konvertibel2-pintu coupéPlatformBMW E64Penyalur dayaMesin3.0 L I6 bensin3.0 L 6 silinder diesel4.4-4.8 L V8 bensinTransmisi6-percepatan manual6-percepatan otomatis6-percepatan (SMG)7-percepatan (SMG)DimensiJarak sumbu roda2.778 mm (109,4 in)Panjang4.831 mm (190,2 in…
Large former industrial site in the city of Essen, North Rhine-Westphalia, Germany Zollverein Coal Mine Industrial Complex in EssenUNESCO World Heritage SiteZollverein Coal Mine, shaft 12LocationEssen, North Rhine-Westphalia, GermanyCriteriaCultural: (ii), (iii)Reference975Inscription2001 (25th Session)Websitewww.zollverein.deCoordinates51°29′29″N 07°02′46″E / 51.49139°N 7.04611°E / 51.49139; 7.04611Location of Zollverein Coal Mine Industrial Complex in G…
Corendon Dutch AirlinesBerkas:Corendon Airlines 2014 logo.svg IATA ICAO Kode panggil CD CND DUTCH CORENDON Didirikan2010Pusat operasiBandar Udara Internasional Schiphol, Bandar Udara Maastricht AachenProgram penumpang setiaCorendonArmada4 (+2 dipesan)Tujuan35Perusahaan indukCorendon Tourism GroupKantor pusatHaarlemmermeer, Holland Utara, BelandarTokoh utamaAtilay Uslu, Jan HeppenerSitus webhttps://www.corendon.com/ Kantor pusat Corendon Dutch Airlines Corendon Dutch Airlines adalah maskapai pene…
Monty Python’s Flying CircusGenreKomedi sketsaKomedi surealSatirKomedi gelapPembuatGraham ChapmanJohn CleeseEric IdleTerry JonesMichael PalinTerry GilliamDitulis oleh Monty Python Neil Innes Douglas Adams Sutradara Ian MacNaughton John Howard Davies PemeranGraham ChapmanJohn Cleese (seri 1-3)Eric IdleTerry JonesMichael PalinTerry GilliamCarol ClevelandLagu pembukaThe Liberty Bell oleh John Philip SousaPenata musikNeil InnesFred Tomlinson SingersNegara asalBritania RayaJmlh. seri4Jmlh. ep…
Juan Demóstenes Arosemena Corregimiento La carretera Panamericana a su paso por Nuevo Arraiján, en el corregimiento de Juan Demóstenes Arosemena. Juan Demóstenes ArosemenaLocalización de Juan Demóstenes Arosemena en Panamá Juan Demóstenes ArosemenaLocalización de Juan Demóstenes Arosemena en Provincia de Panamá OesteCoordenadas 8°55′11″N 79°43′12″O / 8.91972, -79.72Entidad Corregimiento • País Panamá • Provincia Panamá Oeste • Di…
American politician Ryan PearsonMajority Leader of the Rhode Island SenateIncumbentAssumed office January 3, 2023Preceded byMichael McCaffreyMember of the Rhode Island Senatefrom the 19th districtIncumbentAssumed office January 1, 2013Preceded byBethany Moura Personal detailsBorn (1988-06-30) June 30, 1988 (age 35)Providence, Rhode Island, U.S.Political partyDemocraticEducationAmerican UniversityProvidence College (BS)WebsiteCampaign website Ryan W. Pearson (born June 30, 1988) is a…
تمبينكوسي لورش معلومات شخصية الميلاد 22 يوليو 1993 (30 سنة) جنوب إفريقيا الطول 1.67 م (5 قدم 5 1⁄2 بوصة) مركز اللعب وسط الجنسية جنوب إفريقيا معلومات النادي النادي الحالي أورلاندو بيراتس الرقم 3 المسيرة الاحترافية1 سنوات فريق م. (هـ.) 2013–2015 Maluti FET College F.C. [الإنجل…
Rowley Green Common is a six hectare Local Nature Reserve[1][2] and a Site of Importance Metropolitan for Nature Conservation in Arkley, north London.[3][4] It is owned by the London Borough of Barnet. It is also registered common land.[5] It is mainly woodland and heathland, although the most important habitat is the peat bog, one of very few left in London. This hosts star sedge, which is rare in London. The site has hedges at least three hundred years o…
Artikel ini tidak memiliki referensi atau sumber tepercaya sehingga isinya tidak bisa dipastikan. Tolong bantu perbaiki artikel ini dengan menambahkan referensi yang layak. Tulisan tanpa sumber dapat dipertanyakan dan dihapus sewaktu-waktu.Cari sumber: Kerajaan Pucangsula – berita · surat kabar · buku · cendekiawan · JSTOR Kerajaan PucangsulaAbad ke-6–Abad ke-7Ibu kotaDi Lasem, belum dipastikan letak kotaraja-nyaBahasa yang umum digunakanJawa Kuno,…
Lokasi Pengunjung: 3.149.27.81